enter image description hereBriefly, the convolution between $x(nT) e^{-j2\pi f_o nT} e^{-j \pi \Delta_F Tn^2}$ and $c(nT) = e^{j \pi \Delta_F T n^2}$ multiplied $e^{j \pi \Delta_F T k^2}$ allows me to find the DFT $X(f_k = f_o + k\Delta_F)$. Let $N_x$ be the number of samples of $x(nT)$, I found that $c(nT)$ must be at least of length $2N_x$ but I don't know why. Can someone tell me how do I choose the number of samples of $c(nT)$?

  • $\begingroup$ Your question is hard to read. It would help if you write this a complete equation showing all operations and properly defining all your symbols. $\endgroup$
    – Hilmar
    Sep 18 '20 at 11:51
  • $\begingroup$ We need to find the first k-th element of the Fourier transform of $x(nT)$ starting from $f_o$ and going up with steps of amplitude $\Delta_F$. The way to do that is using the following relation $X(f_k = f_o + k\Delta_F) = T \sum_{n=0}^{N_x-1} x(nT) e^{-j2\pi (f_o + k\Delta_F) nT}$ $X(f_k) = T \, e^{-j\pi\Delta_F T k^2} \sum_{n=0}^{N_x-1} x(nT) e^{-j2\pi f_o nT} e^{-j\pi\Delta_F Tn^2} e^{j\pi\Delta_F T (k-n)^2}$ Which can be seen as the convolution of two signals $X(f_k) = e^{-j\pi\Delta_F T k^2} \, T \sum_{n=0}^{N_x-1} z(nT) c(kT-nT)$ The question is $\endgroup$ Sep 18 '20 at 14:35
  • $\begingroup$ How do I properly choose the length of $c(nT)$ ? $\endgroup$ Sep 18 '20 at 14:47
  • $\begingroup$ $T$ is the sampling period $\endgroup$ Sep 18 '20 at 14:57

The convolution in Bluestein's algorithm is a linear convolution, not circular. Since the approach is to use standard FFT's for the convolution operation, the underlying waveforms being convolved must be zero padded out to $M \ge 2N-1$ to compute the linear convolution accurately.

  • $\begingroup$ Yes, $z(nT)$ must be zero padded to perform the convolution but I can also perform the circular convolution without padding $c(nT)$ because $X(f_k)$ can be suppose periodic. For example I can choose $c(nT)$ with $2N_x$ samples pad with $N_x$ the signal $z(nT)$ perform the convolution and discard the first $N_x$ samples. The question is not how to perform the convolution but why do I build the signal $c(nT)$ with $2N_x$ samples and $n$ that goes from $-N_x$ to $N_x-1$ ? Can you explain what "zero padded out" means? Unfortunately I'm not English $\endgroup$ Sep 18 '20 at 23:46
  • $\begingroup$ @AntonioMinighin zero padding means adding zeros before taking the FFT. The point is to do the linear convolution means you can’t assume it is periodic which is what the formula you give also assumes (it is linear convolution not circular) $\endgroup$ Sep 19 '20 at 7:59
  • $\begingroup$ I know what zero padding means but you said "zero padded out", my question was "out" where? At the end of $c(nT)$? Anyway, you can pad $z(nT)$ with $N_x$ values and perform the convolution via dft (circular convolution) and discard the transient (the first $N_x$ samples) the results is right without adding zeros to $c(nT)$, I think this is due to the periodicity of $X(f_k)$. Can you also explain me why do I choose $c(nT)$ with $2N_x$ samples? $\endgroup$ Sep 19 '20 at 9:06

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